Circle

A circle is a simple shape in Euclidean geometry. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius.

A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.

A circle is a plane figure bounded by one line, and such that all right lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre.

Terminology

Circumference: the length of one circuit along the circle, or the distance around the circle.

Diameter: a line segment whose endpoints lie on the circle and which passes through the centre; or the length of such a line segment, which is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord, and it is twice the radius.

Radius: a line segment joining the centre of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.

Sector: a region bounded by two radii and an arc lying between the radii.

Segment: a region, not containing the centre, bounded by a chord and an arc lying between the chord's endpoints.

Secant: an extended chord, a coplanar straight line cutting the circle at two points.

Semicircle: an arc that extends from one of a diameter's endpoints to the other. In non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disk. A half-disk is a special case of a segment, namely the largest one.

Tangent: a coplanar straight line that touches the circle at a single point.

Chord, secant, tangent, radius, and diameter

Arc, sector, and segment

History

The compass in this 13th-century manuscript is a symbol of God's act of Creation. Notice also the circular shape of the halo

The word "circle" derives from the Greekκίρκος/κύκλος (kirkos/kuklos), itself a metathesis of the Homeric Greekκρίκος (krikos), meaning "hoop" or "ring". The origins of the words "circus" and "circuit" are closely related.

The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy, and calculus.

Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.

Some highlights in the history of the circle are:

1700 BCE – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256⁄81 (3.16049...) as an approximate value of π.

Analytic results

Length of circumference

The ratio of a circle's circumference to its diameter is π (pi), an irrationalconstant approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by:

Area enclosed

Area enclosed by a circle = π × area of the shaded square

As proved by Archimedes, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, which comes to π multiplied by the radius squared:

Equivalently, denoting diameter by d,

that is, approximately 79 percent of the circumscribing square (whose side is of length d).

Cartesian coordinates

This equation, known as the Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |x − a| and |y − b|. If the circle is centred at the origin (0, 0), then the equation simplifies to

where t is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the positive x-axis.

An alternative parametrisation of the circle is:

In this parametrisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the line passing through the centre parallel to the x-axis (see Tangent half-angle substitution). However, this parametrisation works only if t is made to range not only through all reals but also to a point at infinity; otherwise, the bottom-most point of the circle would be omitted.

It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane) the points I(1: i: 0) and J(1: −i: 0). These points are called the circular points at infinity.

Polar coordinates

where a is the radius of the circle, is the polar coordinate of a generic point on the circle, and is the polar coordinate of the centre of the circle (i.e., r0 is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x-axis to the line connecting the origin to the centre of the circle). For a circle centred at the origin, i.e. r0 = 0, this reduces to simply r = a. When r0 = a, or when the origin lies on the circle, the equation becomes

In the general case, the equation can be solved for r, giving

Note that without the ± sign, the equation would in some cases describe only half a circle.

Complex plane

In the complex plane, a circle with a centre at c and radius (r) has the equation . In parametric form this can be written .

The slightly generalised equation for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with , since . Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line.

Tangent lines

The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x1, y1) and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x1, y1), so it has the form (x1 − a)x + (y1 – b)y = c. Evaluating at (x1, y1) determines the value of c and the result is that the equation of the tangent is

Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. See circumcircle.

Chord

Chords are equidistant from the centre of a circle if and only if they are equal in length.

The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector are:

An inscribed angle subtended by a diameter is a right angle (see Thales' theorem).

The diameter is the longest chord of the circle.

If the intersection of any two chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then ab = cd.

If the intersection of any two perpendicular chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then a2 + b2 + c2 + d2 equals the square of the diameter.

The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8r2 – 4p2 (where r is the circle's radius and p is the distance from the centre point to the point of intersection).

The distance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.:p.71

Sagitta

The sagitta is the vertical segment.

The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle.

Given the length y of a chord, and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle which will fit around the two lines:

Another proof of this result which relies only on two chord properties given above is as follows. Given a chord of length y and with sagitta of length x, since the sagitta intersects the midpoint of the chord, we know it is part of a diameter of the circle. Since the diameter is twice the radius, the "missing" part of the diameter is (2r − x) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (2r − x)x = (y / 2)2. Solving for r, we find the required result.

Tangent

A line drawn perpendicular to a radius through the end point of the radius lying on the circle is a tangent to the circle.

A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle.

Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.

If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the centre as O, the angles ∠BOA and ∠BPA are supplementary.

If AD is tangent to the circle at A and if AQ is a chord of the circle, then ∠DAQ = 1⁄2arc(AQ).

Theorems

Secant-secant theorem

The chord theorem states that if two chords, CD and EB, intersect at A, then CA × DA = EA × BA.

If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then DC2 = DG × DE. (Tangent-secant theorem.)

If two secants, DG and DE, also cut the circle at H and F respectively, then DH × DG = DF × DE. (Corollary of the tangent-secant theorem.)

The angle between a tangent and chord is equal to one half the subtended angle on the opposite side of the chord (Tangent Chord Angle).

If the angle subtended by the chord at the centre is 90 degrees then l = r √2, where l is the length of the chord and r is the radius of the circle.

If two secants are inscribed in the circle as shown at right, then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs (DE and BC). This is the secant-secant theorem.

Inscribed angles

Inscribed angle theorem

An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding central angle (red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees).

Circle of Apollonius

Apollonius' definition of a circle: d1 / d2 constant

Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant ratio (other than 1) of distances to two fixed foci, A and B. (The set of points where the distances are equal is the perpendicular bisector of A and B, a line.) That circle is sometimes said to be drawn about two points.

The proof is in two parts. First, one must prove that, given two foci A and B and a ratio of distances, any point P satisfying the ratio of distances must fall on a particular circle. Let C be another point, also satisfying the ratio and lying on segment AB. By the angle bisector theorem the line segment PC will bisect the interior angleAPB, since the segments are similar:

Analogously, a line segment PD through some point D on AB extended bisects the corresponding exterior angleBPQ where Q is on AP extended. Since the interior and exterior angles sum to 180 degrees, the angle CPD is exactly 90 degrees, i.e., a right angle. The set of points P such that angle CPD is a right angle forms a circle, of which CD is a diameter.

Second, see:p.15 for a proof that every point on the indicated circle satisfies the given ratio.

Cross-ratios

A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane. If A, B, and C are as above, then the circle of Apollonius for these three points is the collection of points P for which the absolute value of the cross-ratio is equal to one:

Stated another way, P is a point on the circle of Apollonius if and only if the cross-ratio [A,B;C,P] is on the unit circle in the complex plane.

Generalised circles

If C is the midpoint of the segment AB, then the collection of points P satisfying the Apollonius condition

is not a circle, but rather a line.

Thus, if A, B, and C are given distinct points in the plane, then the locus of points P satisfying the above equation is called a "generalised circle." It may either be a true circle or a line. In this sense a line is a generalised circle of infinite radius.

Circles inscribed in or circumscribed about other figures

In every triangle a unique circle, called the incircle, can be inscribed such that it is tangent to each of the three sides of the triangle.

About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three vertices.

A hypocycloid is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle.

Circle as limiting case of other figures

The circle can be viewed as a limiting case of each of various other figures:

A Cartesian oval is a set of points such that a weighted sum of the distances from any of its points to two fixed points (foci) is a constant. An ellipse is the case in which the weights are equal. A circle is an ellipse with an eccentricity of zero, meaning that the two foci coincide with each other as the centre of the circle. A circle is also a different special case of a Cartesian oval in which one of the weights is zero.

A superellipse has an equation of the form for positive a, b, and n. A supercircle has b = a. A circle is the special case of a supercircle in which n = 2.

A Cassini oval is a set of points such that the product of the distances from any of its points to two fixed points is a constant. When the two fixed points coincide, a circle results.

A curve of constant width is a figure whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines. The circle is the simplest example of this type of figure.